Using renormalization group theory on the Ising model using decimation transformations

Question

Consider a 1d Ising model with no external magnetic field $(h=0)$ and adopt a decimation transformation in which every other spin is traced out.
So the Hamiltonian $H$ is given by
$$H = -Jsum_{(i,j)} s_i s_j$$
where $(i,j)$ corresponds to the nearest neighbors of $i$.
I am trying to sketch the RG flow of this system, and I am struggling.
How do you start a problem like this? I know you coarse grain a system by some spin-block tranformation $tau$, but I am unsure of how to use it.
The end goal is to derive the RG equation, so the map $K' = R_l[K]$ for this transformation and find the fixed points in the flow.
Any advice would be appreciated!

QuantumEyedea · Accepted Answer

Take the partition function
$$
Z = sum_{{ S_{j} }} expleft( beta J sum_{( i,j )} s_i s_j right) = sum_{{ s_{j} }} prod_{(i,j)} expleft( beta J s_i s_j right)  ,
$$
and sum over every other spin (so explicitly evaluate ${s_{2n+1}} in pm 1$ in the sum, but don't sum over $s_{2n}$).
This should be equal to a similar partition function over the coarse-grained lattice (with a new coupling $J'$ and new spin variables $s_i'$)
$$
Z = sum_{{ S_{j} }} expleft( beta J' sum_{( i,j )} s'_i s'_j right) = sum_{{ s_{j} }} prod_{(i,j)} expleft( beta J' s'_i s'_j right)  .
$$
By comparing term-by-term, you can relate $J$ to $J'$.

Using renormalization group theory on the Ising model using decimation transformations

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